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Mirrors > Home > ILE Home > Th. List > foeq1 | GIF version |
Description: Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.) |
Ref | Expression |
---|---|
foeq1 | ⊢ (𝐹 = 𝐺 → (𝐹:A–onto→B ↔ 𝐺:A–onto→B)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fneq1 4930 | . . 3 ⊢ (𝐹 = 𝐺 → (𝐹 Fn A ↔ 𝐺 Fn A)) | |
2 | rneq 4504 | . . . 4 ⊢ (𝐹 = 𝐺 → ran 𝐹 = ran 𝐺) | |
3 | 2 | eqeq1d 2045 | . . 3 ⊢ (𝐹 = 𝐺 → (ran 𝐹 = B ↔ ran 𝐺 = B)) |
4 | 1, 3 | anbi12d 442 | . 2 ⊢ (𝐹 = 𝐺 → ((𝐹 Fn A ∧ ran 𝐹 = B) ↔ (𝐺 Fn A ∧ ran 𝐺 = B))) |
5 | df-fo 4851 | . 2 ⊢ (𝐹:A–onto→B ↔ (𝐹 Fn A ∧ ran 𝐹 = B)) | |
6 | df-fo 4851 | . 2 ⊢ (𝐺:A–onto→B ↔ (𝐺 Fn A ∧ ran 𝐺 = B)) | |
7 | 4, 5, 6 | 3bitr4g 212 | 1 ⊢ (𝐹 = 𝐺 → (𝐹:A–onto→B ↔ 𝐺:A–onto→B)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 = wceq 1242 ran crn 4289 Fn wfn 4840 –onto→wfo 4843 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-v 2553 df-un 2916 df-in 2918 df-ss 2925 df-sn 3373 df-pr 3374 df-op 3376 df-br 3756 df-opab 3810 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-rn 4299 df-fun 4847 df-fn 4848 df-fo 4851 |
This theorem is referenced by: f1oeq1 5060 foeq123d 5065 resdif 5091 |
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